The quintic non-homogeneous equation with five unknowns represented by the Diophantine equation () () 3 2 2 4 4 10 P w z y x − = − is analyzed for its patterns of non-zero distinct integral solutions. Various interesting relations between the solutions and special numbers, namely polygonal numbers, pyramidal numbers are exhibited.
Abstract. Non-homogeneous binary quadratic equation represents hyperbola given by 5 6 5 2 2 = − y x is analyzed for its non-zero distinct integer solutions. A few interesting relation between the solution of the given hyperbola, integer solutions for other choices of hyperbola and parabola are obtained.
The hyperbola represented by the binary quadratic equation 23 3 2 2 2 = − y x is analyzed for finding its non-zero distinct integer solutions. A few interesting relations among its solutions are presented. Also, knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are presented. Also, employing the solutions of the given equation, special Pythagorean triangle is constructed.
The binary quadratic equation represents by negative Pellian ݕ ଶ = ݔ04 ଶ − 15 is analyzed for its distinct integer solutions.A few interesting relations among the solution are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas, parabolas and Pythagorean triangle.
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